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Properties

Label 2-600-120.59-c1-0-20
Degree $2$
Conductor $600$
Sign $-0.667 - 0.744i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.15 + 0.814i)2-s + (0.887 + 1.48i)3-s + (0.672 + 1.88i)4-s + (−0.185 + 2.44i)6-s − 0.797·7-s + (−0.757 + 2.72i)8-s + (−1.42 + 2.64i)9-s − 0.320i·11-s + (−2.20 + 2.67i)12-s − 4.30·13-s + (−0.921 − 0.649i)14-s + (−3.09 + 2.53i)16-s + 2.57·17-s + (−3.79 + 1.89i)18-s + 6.10·19-s + ⋯
L(s)  = 1  + (0.817 + 0.576i)2-s + (0.512 + 0.858i)3-s + (0.336 + 0.941i)4-s + (−0.0756 + 0.997i)6-s − 0.301·7-s + (−0.267 + 0.963i)8-s + (−0.474 + 0.880i)9-s − 0.0966i·11-s + (−0.636 + 0.771i)12-s − 1.19·13-s + (−0.246 − 0.173i)14-s + (−0.773 + 0.633i)16-s + 0.624·17-s + (−0.894 + 0.446i)18-s + 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01791 + 2.27950i\)
\(L(\frac12)\) \(\approx\) \(1.01791 + 2.27950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.814i)T \)
3 \( 1 + (-0.887 - 1.48i)T \)
5 \( 1 \)
good7 \( 1 + 0.797T + 7T^{2} \)
11 \( 1 + 0.320iT - 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 - 6.10T + 19T^{2} \)
23 \( 1 - 3.13iT - 23T^{2} \)
29 \( 1 - 8.79T + 29T^{2} \)
31 \( 1 + 9.90iT - 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
41 \( 1 - 5.28iT - 41T^{2} \)
43 \( 1 + 2.97iT - 43T^{2} \)
47 \( 1 - 6.56iT - 47T^{2} \)
53 \( 1 - 3.94iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + 8.83iT - 61T^{2} \)
67 \( 1 + 4.66iT - 67T^{2} \)
71 \( 1 - 3.43T + 71T^{2} \)
73 \( 1 + 1.43iT - 73T^{2} \)
79 \( 1 + 2.89iT - 79T^{2} \)
83 \( 1 + 3.37T + 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 4.26iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.14037479267061195849884112442, −9.797307938794907354484333264954, −9.449208221839209682024781357980, −8.003415596084526491812302866722, −7.63909893744639181748150956413, −6.30925199472315275052324032061, −5.27763768217896596358209943533, −4.55087223942103150521354471993, −3.40465926774775496944177209890, −2.59571861270832167002548658286, 1.07823822080919812320994713343, 2.55895087809225747821236549116, 3.26269032285380597515078089452, 4.67834444214834940352045351723, 5.69752017204570888892069629720, 6.77939733774011557567216845658, 7.41071721256271368465415189596, 8.631466258817443900037086029676, 9.705546070671424794040683123946, 10.27604900624104249333266775450

Graph of the $Z$-function along the critical line